For functions \(u:M\rightarrow \mathbb {R}\) , the Lebesgue and
Sobolev spaces \(L^p(M)\)  and \(W^{1,p}(M)\)  are defined as in [1]}.
The dual space of \(W^{1,p}(M)\)  is
\(W^{-1,p}(M):=\left(W^{1,q}(M)\right)^{\prime },\,\,\,\,\,\,\,\frac{1}{p}+\frac{1}{q}= 1.\) 
